Bar model method: Singapore maths model drawing with worked examples

The bar model method is the way Singapore primary school pupils solve maths word problems by drawing each quantity as a bar. Children in Singapore learn it from Primary 1, starting with part-whole and comparison bars, and use it up to the PSLE for fractions, percentage, ratio and before-after questions. It was created in 1983 at the Curriculum Development Institute of Singapore and is the "pictorial" step of MOE's Concrete-Pictorial-Abstract approach.

Below: the Singapore math method, the six main bar models with worked examples, which level learns which model from P1 to P6, the link to Sec 1 algebra, common mistakes and how to help at home. Facts are from MOE, SEAB, the National Institute of Education (NIE) and the official TIMSS and PISA reports, checked as of 7 Nov 2026.

What is the Singapore math method?

The Singapore math method is the way maths is taught in Singapore's primary schools under MOE's syllabus. It puts mathematical problem solving at the centre, supported by five parts: concepts, skills, processes, metacognition and attitudes. Lessons move from concrete objects to pictures to symbols, and the bar model is the best known of those pictures.

MOE's 2021 Primary Mathematics Syllabus, which covers Primary 1 to Primary 6, says it plainly: "The central focus of the mathematics curriculum is the development of mathematical problem-solving competency." The framework is drawn as a pentagon. Mathematical Problem Solving sits in the middle and the five supporting parts form the sides.

Source: MOE, 2021 Primary Mathematics Syllabus, updated Oct 2025, page 13.

The syllabus also names Pólya's four steps to problem solving and the use of heuristics as general strategies for non-routine problems. NIE's Kaur calls the model method a key heuristic for whole-number word problems.

Concrete-Pictorial-Abstract (CPA) is the teaching approach that goes with the framework. The official TIMSS 2015 encyclopedia entry for Singapore describes it as central to MOE's pedagogy at primary and lower secondary level. Children first handle real objects (concrete), then draw them (pictorial), then work with numbers and symbols alone (abstract). MOE's 2013 syllabus, quoted in a 2015 NIE paper, describes the teacher as a facilitator who "guides students through the concrete, pictorial and abstract levels of understanding".

The 2021 syllabus adds three focus areas: building mathematical processes for 21st century competencies, a greater awareness of "big ideas" such as equivalence and proportionality, and more attention to metacognition.

Where did the Singapore math method come from?

The Singapore math method grew out of a 1979 to 1983 national effort to raise primary maths standards. The Curriculum Development Institute of Singapore (CDIS) opened in June 1980, its Primary Mathematics Project produced the first primary maths curriculum in 1981 using the CPA approach, and in 1983 the team, led by Dr Kho Tek Hong, introduced the model method for word problems.

NIE researchers give this history. Professor Berinderjeet Kaur writes that low achievement in maths was a concern in the late 1970s, so MOE asked CDIS to write the syllabuses, textbooks, workbooks and teacher guides. In 1983 the team "made a breakthrough" on the difficulties pupils had with word problems. The method first went into the curriculum for Primary 5 and 6 in the late 1980s, and it is now taught from Primary 1.

Sources: Kaur (2015), The model method, NIE repository. Leong, Ho and Cheng (2015), Concrete-Pictorial-Abstract, The Mathematics Educator. Lee (2010), curriculum frameworks paper, NIE repository.

What is the bar model method?

The bar model method, also called model drawing or the Singapore bar model, is a way of solving word problems by drawing each quantity as a rectangle. The length of each bar stands for the size of the quantity. Once the bars are drawn and labelled, the link between the known numbers and the unknown becomes visible, so the child can see whether to add, subtract, multiply or divide.

Kaur explains why rectangles were chosen: they are easier to cut into smaller equal units than other shapes. That matters because most upper primary questions are solved by finding the value of one unit.

Two points are worth knowing before the worked examples:

• A bar model is a picture of the structure, not the answer. England's National Centre for Excellence in the Teaching of Mathematics (NCETM) puts it this way: the bar model "is not a method for solving problems" on its own, but a way of revealing the structure inside one. The child still has to calculate.
• Bars only need to be roughly to scale. A bar for 40 should look longer than a bar for 25. Exact lengths are not needed, but equal units must look equal.

In the US, the same picture is called a tape diagram. It is named in the Common Core standards for Grade 6 ratio reasoning.

How do you draw a bar model, step by step?

To draw a bar model, read the whole question, decide who or what is being compared, draw one bar for each, and mark the known numbers and the unknown with a question mark. Then find the value of one unit and build the answer from it. Most PSLE-style questions follow the same eight steps.

What are the main types of bar model?

There are six bar models most Singapore children use: part-whole, comparison, multiplication and division, fractions, ratio and before-after. Part-whole and comparison are the two basic shapes that NIE describes, and the other four build on them. The table shows each model, the question it answers and where it first appears in the MOE syllabus.

Source for the topics: MOE, 2021 Primary Mathematics Syllabus. The syllabus lists topics, not model names. The last column is the first topic where each model is usually drawn.

How does the part-whole model work, including for subtraction?

The part-whole model shows a whole made of two or more parts. If the parts are known, add them to find the whole. If the whole and one part are known, subtract to find the missing part. That is why the part-whole bar is the main bar model for subtraction: the child sees the missing piece instead of guessing which number to take away.

Worked example 1 (P1 to P2, addition). Mei has 18 star stickers and 17 heart stickers. How many stickers does she have altogether?

The whole is 18 + 17 = 35. Mei has 35 stickers altogether.

Worked example 2 (P2 to P3, subtraction). A shop had 524 eggs. After selling some, it had 187 left. How many eggs were sold?

The missing part is 524 − 187 = 337. Check: 337 + 187 = 524. The shop sold 337 eggs.

A part-whole bar can have more than two parts. In P3 a child might split 1,000 into three parts and find the third by subtracting the other two from the whole.

How does the comparison model work?

The comparison model puts two or more bars one above the other, lined up on the left, so the difference between them shows as an extra piece on the longer bar. It answers "how many more", "how many fewer" and "how many altogether" when one amount is described through another. The extra piece is the difference.

Worked example 3 (P2 to P3). Ali has 46 marbles. Ben has 19 more marbles than Ali. How many marbles do they have altogether?

Ben has 46 + 19 = 65 marbles. Together they have 46 + 65 = 111 marbles.

A common slip here is to stop at 65, which answers "how many does Ben have", not the question asked. Step 7 in the drawing routine, answering the question asked, is the fix.

How do bar models show multiplication and division?

For multiplication, the bar model uses equal units: if one amount is 3 times another, its bar is 3 units long and the other bar is 1 unit. For division, one bar is cut into equal parts, either to share a total among groups or to find how many groups fit. Finding the value of one unit is the key step in both.

Worked example 4 (P3, "times as many"). Sara has 3 times as many beads as Lin. Together they have 84 beads. How many beads does Sara have?

There are 4 units in all. 4 units = 84, so 1 unit = 84 ÷ 4 = 21. Sara has 3 × 21 = 63 beads. Check: Lin has 21, and 63 + 21 = 84.

Worked example 5 (P2 to P3, sharing). $96 is shared equally among 8 children. How much does each child get?

One part is $96 ÷ 8 = $12. Each child gets $12.

How do bar models work for fractions?

For fractions, the bar stands for one whole and is cut into as many equal units as the denominator. A question such as "2/5 of his money" means 2 of 5 equal units. When a question says "of the remainder", the child redraws or recuts the remaining part. This is where bar models save the most time in P5.

Worked example 6 (P5). Jun spent 2/5 of his money on a book and 1/3 of the remainder on a pen. He had $24 left. How much money did he have at first?

The book takes 2 of 5 units, leaving 3 units. One third of those 3 units is 1 unit for the pen, leaving 2 units. 2 units = $24, so 1 unit = $12. At first Jun had 5 × $12 = $60. Check: book $24, remainder $36, pen $12, left $24.

Percentage works the same way. A bar for 100% can be cut into 4 units of 25% or 5 units of 20%, and NIE's Kaur shows a before-after percentage question solved exactly this way.

Our PSLE heuristics guide has a further fraction example worked as a full PSLE-style answer.

How do bar models solve ratio questions?

A ratio bar model draws each quantity as a number of equal units matching its ratio term. If Amy's stamps to Ben's stamps is 3 : 5, Amy's bar has 3 units and Ben's has 5 units of the same size. Any fact in the question, such as a difference or total, then tells the child what a set of units is worth.

Worked example 7 (P6). The ratio of Amy's stamps to Ben's stamps is 3 : 5. Ben has 28 more stamps than Amy. How many stamps do they have altogether?

Ben's bar is 2 units longer, and those 2 units = 28 stamps. So 1 unit = 14. Altogether there are 3 + 5 = 8 units, which is 8 × 14 = 112 stamps.

Ratio of three quantities, such as a : b : c, uses three bars the same way. Ratio is a P6 topic in the 2021 syllabus, which also covers equivalent ratios and the link between fractions and ratio. Foundation Mathematics does not include ratio.

How do before-after bar models work?

A before-after model, also called a change model, draws the situation twice: once before something is given, spent or added, and once after. The child looks for what stays the same, such as the total, the difference or one person's amount, and uses it to link the two drawings. NIE notes that P5 and P6 pupils use part-whole and comparison models with this before-after idea.

Worked example 8 (P5, total unchanged). Kim and Raj had $120 altogether. After Kim gave Raj $15, they had the same amount. How much did Kim have at first?

Money only moved between them, so the total stays at $120. After, each has $120 ÷ 2 = $60. Before giving $15, Kim had $75.

Worked example 9 (P6, one amount unchanged). The ratio of Nora's beads to Omar's beads was 4 : 3. After Omar bought 40 more beads, the ratio became 4 : 5. How many beads did Omar have at first?

Nora's beads did not change, so her 4 units stay 4 units and the units are the same size before and after. Omar went from 3 units to 5 units, so 2 units = 40 beads and 1 unit = 20. Omar had 3 × 20 = 60 beads at first. Check: Nora has 80, and 80 : 100 is 4 : 5.

Age questions use a third idea: the difference stays the same. A mother aged 36 and a son aged 8 are always 28 years apart. When she is 3 times his age, the difference is 2 units, so 1 unit = 14 and the son is 14. That happens in 6 years.

Which level learns which bar model, from P1 to P6?

Children in Singapore meet part-whole and comparison bars in P1 and P2, use "times as many" and division bars in P2 and P3, add fraction bars in P2 to P4, and combine everything with percentage, rate, ratio and before-after changes in P5 and P6. The order follows when each topic appears in MOE's 2021 Primary Mathematics Syllabus.

Source: MOE, 2021 Primary Mathematics Syllabus, Content by Level, pages 31 to 43.

Three points for 2026, 2027 and later cohorts:

• The 2021 syllabus now covers every level. MOE's document says it reached Primary 6 from 2026, so the 2026 PSLE cohort was the first to sit the PSLE under it. The 2027 cohort and those after follow the same syllabus, as of 7 Nov 2026.
• P1 to P4 is common to all pupils. From P5, children take Standard or Foundation Mathematics. MOE says Foundation revisits key P1 to P4 concepts, and its new content is a subset of Standard. Foundation has no ratio or algebra strand. Our Foundation vs Standard guide explains the choice.
• Schools choose their own textbooks and pacing. The syllabus fixes what is learned by the end of each year. A school may teach a model a term earlier or later than a friend's school.

How do bar models fit PSLE heuristics and the PSLE Maths paper?

Drawing a model is one of the problem-solving heuristics MOE's syllabus asks teachers to build, alongside other strategies such as making a list or working backwards. In the PSLE, it is most useful in the longer Paper 2 word problems on fractions, percentage, ratio and before-after changes, where the bar shows the structure and the working.

SEAB's PSLE Mathematics paper (syllabus 0008) has two papers worth 100 marks. Paper 1 is 1 hour 10 minutes without a calculator and Paper 2 is 1 hour 20 minutes with a calculator (SEAB, PSLE Mathematics syllabus 2026). As of 7 Nov 2026, SEAB had not posted a 2027 format page, so check SEAB's site for the 2027 papers.

This guide keeps heuristics short on purpose. Our PSLE Math heuristics guide covers each heuristic, when to use it and how method marks work. For a wider plan, see how to score AL1 in PSLE Math.

How does the bar model lead to algebra in Sec 1?

A bar model unit is an unknown waiting for a letter. When a P6 child writes "4 units = 84", a Sec 1 student writes "4x = 84". MOE's P6 syllabus starts algebra with "using a letter to represent an unknown number" and simple linear equations, and the Sec 1 syllabus asks students to form a linear equation in one variable to solve problems.

Worked example 10 (P6 and Sec 1). A pen costs $3 more than a pencil. 2 pens and 3 pencils cost $16. Find the cost of a pen.

Check: 2 × $5 + 3 × $2 = $16.

The bar is not dropped in Sec 1. It becomes the bridge: a student can sketch the bar first, then write the equation from it. By Sec 2, MOE's G3 syllabus adds simultaneous linear equations in two variables, which is where many of the hardest PSLE bar questions turn into two-line algebra.

Source: MOE, G2 and G3 Mathematics syllabuses (2020 S1 cohort), Secondary One N5 and N7 and Secondary Two N7. Our Secondary 1 guide covers the jump in more detail.

What mistakes do children make with bar models?

The most common bar model mistakes are drawing before reading the whole question, making equal units look unequal, labelling the wrong bar, mixing up "more than" with "times as many" and forgetting to answer the actual question. Most are fixed by slowing down at steps 1, 5 and 7 of the drawing routine rather than by more practice papers.

How can parents help with bar models at home?

Parents help most by asking the child to explain the drawing aloud, keeping numbers small at first and following the school's way of drawing. Start with real objects for younger children, move to drawn bars, then ask for the number sentence. Short, regular practice on one model type at a time works better than mixed papers too early.

What not to do: do not teach algebra shortcuts to replace bars in P3 to P5 unless the school does. Children may be asked to show working that the teacher can follow, and a jump to algebra can confuse rather than help. If you were taught differently, the school's parent briefing slides or textbook examples are the safest model to follow.

Is the Singapore math method used in other countries?

Yes. Schools in England, the United States and elsewhere use Singapore-style bar models. England's National Centre for Excellence in the Teaching of Mathematics publishes bar model guidance for primary teachers, and the US Common Core standards name "tape diagrams" for ratio reasoning in Grade 6. Singapore students have scored at or near the top of recent international studies.

Sources: TIMSS 2023 Grade 4 maths achievement, TIMSS 2023 Grade 8 maths achievement, MOE on TIMSS 2023, MOE on PISA 2022, MOE on PISA 2025.

Two cautions. MOE noted that P4 maths in TIMSS 2023 dipped slightly from 2019, and fewer P4 pupils said they enjoy maths "a lot" (42%, down from 48%). Scores also reflect the whole system, including teachers, textbooks and home support, not the bar model alone.

When does a tutor help with bar models?

A tutor helps most when a child can do the calculations but freezes at word problems, draws bars that do not match the story, or reaches P5 still unsure of part-whole and comparison models. One-to-one time lets the tutor watch the child draw, spot the exact step that goes wrong and rebuild from there, in the school's own style.

Signs worth acting on: marks lost mainly in Paper 2 word problems, or the same "more than" and "times" mix-up across several tests. Our guides on when to start tuition and how many hours of tuition before the PSLE help with timing.

Key terms

• Bar model. A drawing of rectangles whose lengths stand for the quantities in a word problem.
• Model drawing. Another name for the bar model method, used in many Singapore schools.
• Unit. One equal part of a bar. Most questions are solved by finding what 1 unit is worth.
• Part-whole model. A bar split into parts that together make the whole.
• Comparison model. Two or more bars lined up so their difference can be seen.
• Before-after model. Two drawings of the same situation, before and after a change.
• CPA. Concrete-Pictorial-Abstract, MOE's approach of moving from objects to pictures to symbols.
• Heuristic. A general problem-solving strategy, such as drawing a model or working backwards.
• Mathematics Curriculum Framework. MOE's pentagon with mathematical problem solving at the centre and concepts, skills, processes, metacognition and attitudes around it.
• Tape diagram. The US name for a bar model.